Weight Summability of Solutions of the Sturm–Liouville Equation

We consider an equation−y″(x)+q(x) y(x)=f(x), x R; ((1))wheref(x) Lp(R),p [1, ∞] (L∞(R) C(R)), 1 q(x) Lloc1(R). We study requirements for a weight functionr(x) Llocp(R) and forq(x) under which, for a givenp [1, ∞], regardless off(x) Lp(R), the solutiony(x) Lp(R) of Eq. (1) satisfies the inequalities: r(x) y(x) p c f(x) p, y″(x) p+ q(x) y(x) p c f(x) p, c=const.